Magnetic Forces: Intro

Magnetic Forces: Introduction

We now consider the forces on moving test charges due to currents.  What we normally mean by a current is a wire that has charges moving along its length, and counter charges that are stationary and balance most, if not all, the electrostatic field from the moving charges.  In a real wire with current, there is a stationary matrix of positive charges and a moving fluid of negative charges.  The positive charges are the nuclei of all the atoms of the conductor, reduced by the negatively charged bound electrons.  The moving negative charge is the collective motion of all the conduction band electrons.  To a very good approximation, the positive and negative charge densities sum to zero in the observers frame. 

In the calculations that follow, we will be dealing with three different reference frames: the source charge rest frame, the test charge rest frame, and the observers frame.  There are two source charge frames in this problem, one for the (stationary) positive charges and another for the moving negative charges.  The test charge, whose motion we are trying to understand, is generally moving with respect to one or more of the source charges.  We have already calculated in the earlier sections the forces on a stationary test charge.  We will build on that understanding to calculate how the test charge moves in the frame of an observer  that is not at rest with the test charge.

With three different reference frames, connected by the Lorentz Transformation, it can get very confusing which frame which parameter is based in.  To try to keep this all straight we will follow a strict nomenclature of subscripts:

-All parameters in the source charge frame will have a capital S subscript: e.g. x_s, rho_x etc.  These subscripts may in addition have a + or – to indicate the polarity of the source charge.

- All parameters in the test charge frame will have a capital T subscript: e.g. x_T, rho_T etc.

- All parameters in the observers frame will have a capital O subscript: e.g. x_O, rho_O etc.

-In addition, all parameters that relate to motion between frames will have two subscripts indicating first the reference frame in which the parameter is observed, and second the reference frame it refers to.  For example, the parmeter gamma_OT is the Lorentz Transform gamma parameter used to transform from the observers coordinates to the test charge coordinates.  Likewise beta_TS is the velocity of the source charge as seen from the test charge frame.

We can now construct the picture of this problem in these three frames, with all the parameters we will need defined in each frame:

Observers Frame:  We will place the origin of all the frames at the test charge.  In the observers frame the wire with the current is parallel to the Y axis and crosses the X axis at position x_O and z_O = 0.  Thus the test charge is momentarily at a radius r_O = x_O from the wire. 

We will be considering two cases: one with the test charge moving along the Y axis, parallel to the wire, and another case where the test charge is moving toward/away from the wire.  We will also briefly consider the test charge moving in the Z direction, just to show that there is no magnetic force resulting from that motion.  The velocity of the test charge is called beta_OT in this frame, and has a corresponding gamma_OT.

The source charges have velocities beta_OS+ and beta_OS-, with associated gamma parameters gamma_OS+ and gamma_OS+.  Likewise the ordinary velocity of the test charge is v_OT.

Since the positive charges are stationary in the observers frame, the density of positive charges in their own frame is the same as that in the observers frame:

The negative charges are moving, but as seen in this frame have the same density as the positive charges, but the opposite sign:

The test charge we will simply call q_T, and is the same in all frames.

Source Charge Frame:  There are two source charge frames, one for the positive charges and one for the negative charges.  For simplicity, and to understand the most typical situations, we will assume the positive charges are stationary in the observers frame.  The negative charges are moving along a line parallel to the Y axis.  In the negative source charge frame the observer frame is moving with a velocity:

And

The density of positive charges in their own frame is the same as that in the observers frame:

The negative charges have a density in the observers frame of  which we assume is equal to the positive charge density:

The test charge is also moving at a velocity  with Lorentz Transform factor  that we will calculate for each case.

Test Charge Frame:  From the test charge frame, the velocity of the observer is:

And

The charge densities of the positive and negative source charges are, in general, different from the densities in the observers frame.  We will have to calculate these for each problem.  We will call them rho_T+ and rho_T-.  The motion of the source charges also will have to be calculated for each problem.  They will be called beta_TS+ and beta_TS-.  The corresponding gamma parameters are gamma_TS+ and gamma_TS-.

One final introductory note concerning the speeds of the charges in a current carrying wire:  The density of charges in typical wires (e.g. copper) is extremely high, so even at very high currents the charges are actually moving quite slowly.  For example in copper conducting current at the maximum limit that copper is capable of maintaining continuously, the speed of the electrons is about 1 mm/second.  This shows that normal magnetic fields are the result of charges moving at non-relativistic speeds.  In our calculations we will not restrict ourselves to only non-relativistic moving charges.  Our result will be completely general.